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direct observation of Figure 2.22a we obtain the following characteristic tran-
sient rheological responses: path 1: evolution from velocity direction (
θ
=
0 ,
θ
0), the shear stress, will present one overshoot (path denoted by 1); path
2: evolution from velocity gradient direction (
=
0), the shear stress
will increase monotonically; path 3: evolution from vorticity direction (
θ
=
90,
θ
=
θ
=
0 ,
θ
90); the shear stress will present an undershoot. Figure 2.22b shows that
the shear stress landscape is dense with local maxima and local minima and
that monotonic responses are more the exception than the rule. Anisotropic
effects introduced by director stress coupling do provide an important mecha-
nism of stress nonmonotonicity in transient shear startup fl ows. The viscosity
maxima, minima, and their difference in the fi gure are:
=
=++
−
−
αααα
αα
ααα α
+
+
4
1
5
6
3
2
3
6
η
−
2
(2.58)
max
22
2
4
3
2
=+
−
−
ααα
αα
αα
4
5
6
3
6
η
+
(2.59)
min
2
2
2
3
2
The viscosity jump between these extreme,
Δη
=
η
max
−
η
min
, is given by
(
)
=−
−
ααα
αα
αα
+
1
5
6
2
3
∆η
(2.60)
4
−
4
3
2
Equations (2.58)-(2.60) have been used to assess the degree of the accuracy
of the planar transversal isotropic fl uid, and of the nonplanar Leslie-Ericksen
equations, by comparing them with experimental measures (Ericksen, 1969;
Han and Rey, 1994a - c ; Rey, 1993a,b ).
2.4.1.6 Scaling Ericksen Theory
Ericksen (1969) has shown that there
are certain scaling properties of the LE theory that are useful for rheological
characterization. Below we present the results of Ericksen's scaling work,
which is used in the discussions of results. For a fi xed material constant set,
the viscosity
η
scaling for simple shear fl ow is given by
(
)
=
(
)
EE
=
or
ηγ
h
2
ηγ
h
2
if
EE
=
or
γ
h
2
=
γ
h
2
(2.61)
α
β
α
α
β
β
α
β
α
α
β
β
denote two different conditions. Ericksen's scaling also shows
that time (
t
) for different plate spacing
h
scales with
h
2
, in addition to the
length scaling with
h
. Equation (2.61) may be more general by including the
time scaling, for a different combination of shear rate
where
α
and
β
γ
and plate spacing
h
,
as follows:
(
)
=
(
)
ηγ
ht
2
,
ηγ
ht
2
,
(2.62)
αα α
ββ β
where the relationship between different times can be readily shown to be
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